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QUIZ 3
STAT4351;
Name (Please Print):
1. Consider a binomial random variable Binom(θ, n). Let p(x|θ, n) deP
notes its probability mass function. Use nx=0 p(x|θ, n) = 1, differentiate both
sides with respect to θ, and then use the result to prove that the expectation
of X is nθ. Show all steps of the proof.
2. For the binomial random variable in Problem 1, prove that
p(x + 1|θ, n) = θ(n − x)[(x + 1)(1 − θ)]−1 p(x|θ, n).
3. Moment generating function uniquely defines exponential and Gamma
random variables. Using this fact, prove that the sum of independent and
identically distributed exponential random variables has a Gamma distribution. Write down the proof and the corresponding Gamma distribution.
4. A manufacturer claims that poor quality of supplies yields 70% of
defected TVs. Find the probability that 5 of the next 6 TVs will be defected
due to the poor quality of supplies. Use both the formula and Table.
5. Among the 300 employees of a company, 240 are union members. If six
of the employees are chosen to serve on a committee, what is the probability
that four of the six will be union members?
6.If 18 defective glass bricks include 10 that have cracks but no discoloration, five that have discoloration but no cracks, and three that have cracks
and discoloration, what is the probability that among six of the bricks (chosen at random for further checks) three will have cracks but no discoloration,
one will have discoloration but no cracks, and two will have cracks and discoloration?
7. Consider an exponential random variable with parameter θ. (a) Find
the probability that it will take on value smaller than −θ ln(1 − p). (b) Find
its hazard function f (x)/(1 − F (x)) where f is the density and F is the cdf.
β
8. Consider a probability density function fX (x) = kxβ−1 e−αx I(x > 0).
Find k and then the expected value.
9. Let X and Y have a bivariate normal distribution. Find the correlation
coefficient between U = 2X + Y and V = X + 2Y .
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